Admissible observation operators for linear semigroups

Consider a semigroupT on a Banach spaceX and a (possibly unbounded) operatorC densely defined inX, with values in another Banach space. We give some necessary as well as some sufficient conditions forC to be an admissible observation operator forT, i.e., any finite segment of the output functiony(t)=CTtx,t≧0, should be inLp and should depend continuously on the initial statex. Our approach is to start from a description of the map which takes initial states into output functions in terms of a functional equation. We also introduce an extension ofC which permits a pointwise interpretation ofy(t)=CTtx, even if the trajectory ofx is not in the domain ofC.

[1]  D. L. Russell,et al.  Admissible Input Elements for Systems in Hilbert Space and a Carleson Measure Criterion , 1983 .

[2]  Stuart Townley,et al.  A stability radius for infinite dimensional systems , 1987 .

[3]  P. Fuhrmann.  Linear Systems and Operators in Hilbert Space , 1982 .

[4]  David L. Russell,et al.  A General Theory of Observation and Control , 1977 .

[5]  Harry Dym,et al.  Fourier series and integrals , 1972 .

[6]  D. Salamon.  Infinite Dimensional Linear Systems with Unbounded Control and Observation: A Functional Analytic Approach. , 1987 .

[7]  George Weiss,et al.  The representation of regular linear systems on Hilbert spaces , 1989 .

[8]  R. Triggiani,et al.  Stabilization of Neumann boundary feedback of parabolic equations: The case of trace in the feedback loop , 1983 .

[9]  J. Goldstein.  Semigroups of Linear Operators and Applications , 1985 .

[10]  Observation and Prediction for One-Dimensional Diffusion Equations , 1975 .

[11]  George Weiss.  Admissibility in input elements for diagonal semigroups on l 2 , 1988 .

[12]  George Weiss,et al.  Admissibility of unbounded control operators , 1989 .

[13]  R. Curtain.  Infinite-Dimensional Linear Systems Theory , 1978 .

[14]  R. Curtain.  On semigroup formulations of unbounded observations and control action for distributed systems , 1984 .

[15]  A. J. Pritchard,et al.  The linear quadratic control problem for infinite dimensional systems with unbounded input and outpu , 1987 .

[16]  A. J. Pritchard,et al.  Unbounded Control and Observation Systems and Their Duality , 1978 .

[17]  Dietmar A. Salamon,et al.  On control and observation of neutral systems , 1982 .

[18]  Ruth F. Curtain,et al.  Well posedness of triples of operators (in the sense of linear systems theory) , 1989 .

[19]  R. Nagel,et al.  One-parameter Semigroups of Positive Operators , 1986 .